Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldm1cossres2 Structured version   Visualization version   GIF version

Theorem eldm1cossres2 35819
Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
Assertion
Ref Expression
eldm1cossres2 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉

Proof of Theorem eldm1cossres2
StepHypRef Expression
1 eldm1cossres 35818 . 2 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝑥𝑅𝐵))
2 elecALTV 35645 . . . 4 ((𝑥 ∈ V ∧ 𝐵𝑉) → (𝐵 ∈ [𝑥]𝑅𝑥𝑅𝐵))
32el2v1 35608 . . 3 (𝐵𝑉 → (𝐵 ∈ [𝑥]𝑅𝑥𝑅𝐵))
43rexbidv 3283 . 2 (𝐵𝑉 → (∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅 ↔ ∃𝑥𝐴 𝑥𝑅𝐵))
51, 4bitr4d 285 1 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  wrex 3131  Vcvv 3469   class class class wbr 5042  dom cdm 5532  cres 5534  [cec 8274  ccoss 35571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ec 8278  df-coss 35777
This theorem is referenced by:  eldmqs1cossres  36011
  Copyright terms: Public domain W3C validator